Properties

Degree 1
Conductor $ 3 \cdot 7 $
Sign $-0.0633 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0633 - 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $-0.0633 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{21} (11, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 21,\ (1:\ ),\ -0.0633 - 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.119373288 - 1.192660627i$
$L(\frac12,\chi)$  $\approx$  $1.119373288 - 1.192660627i$
$L(\chi,1)$  $\approx$  1.141608742 - 0.7593774607i
$L(1,\chi)$  $\approx$  1.141608742 - 0.7593774607i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−40.23685401838672866861432820238, −38.567990239189238876830991233428, −37.14319148227260714163461378756, −35.5280025520395201455620714587, −34.36460321311181453481080963285, −33.31808790813385758521621654512, −32.140109237214601411081419729094, −30.6363221120543385600721457223, −29.59888903169292550313337820430, −27.40042617941045636858938267248, −26.09555313814283836834102187054, −25.09182551808548267340125569260, −23.54291842636638356773863602664, −22.29173374741619256524879198176, −21.17602198380544411305641449277, −18.78076149692662465730077844967, −17.45313784141581656464295650630, −15.92053869855272052388183740450, −14.41833423973691722484914262443, −13.35979207058571223825215003508, −11.25009364810249046204992106384, −9.02729596205879432665264889066, −7.08050476257602177700061314530, −5.70086521677533954748123778311, −3.38123798130451827202973386396, 1.61201939105444709858788997003, 4.13202204632408647064707313865, 5.8751261712862967192297656850, 8.84449363843428597306641234141, 10.329513358619587930526372948162, 12.17774973539419563968302301124, 13.28402633710605058592580067155, 14.87263645319252306374596971776, 16.95697615757441970894706927005, 18.6285197154408709494386107626, 20.26899064118222038336848795427, 21.12402284421267552356689866092, 22.666672209770741748167873251, 24.00718281745723516950085479751, 25.5147724933501927559619647108, 27.69053131480802296539058343815, 28.5029282750993406787087103555, 29.863580418798960306799522844544, 31.10642269186750944554083916492, 32.484204295731009453098836670545, 33.32383167552013224190719473763, 35.613497094931050093311998530581, 36.697864474498195969919747577484, 37.93560052427407836388259041719, 39.189747011792309352002046788789

Graph of the $Z$-function along the critical line